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We examine the composition of the L∞ norm with weakly convergent sequences of gradient fields associated with the homogenization of second order divergence form partial differential equations with measurable coefficients. Here the sequences of coefficients are chosen to model heterogeneous media and are piecewise constant and highly oscillatory. We identify local representation formulas that in the fine phase limit provide upper bounds on the limit superior of the L∞ norms of gradient fields. The local representation formulas are expressed in terms of the weak limit of the gradient fields and local corrector problems. The upper bounds may diverge according to the presence of rough interfaces. We also consider the fine phase limits for layered microstructures and for sufficiently smooth periodic microstructures. For these cases we are able to provide explicit local formulas for the limit of the L∞ norms of the associated sequence of gradient fields. Local representation formulas for lower bounds are obtained for fields corresponding to continuously graded periodic microstructures as well as for general sequences of oscillatory coefficients. The representation formulas are applied to problems of optimal material design.
@article{M2AN_2012__46_5_1121_0,
author = {Lipton, Robert and Mengesha, Tadele},
title = {Representation formulas for $L^\infty $ norms of weakly convergent sequences of gradient fields in homogenization},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1121--1146},
publisher = {EDP-Sciences},
volume = {46},
number = {5},
year = {2012},
doi = {10.1051/m2an/2011049},
zbl = {1273.35038},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2011049/}
}
TY - JOUR AU - Lipton, Robert AU - Mengesha, Tadele TI - Representation formulas for $L^\infty $ norms of weakly convergent sequences of gradient fields in homogenization JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 1121 EP - 1146 VL - 46 IS - 5 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2011049/ DO - 10.1051/m2an/2011049 LA - en ID - M2AN_2012__46_5_1121_0 ER -
%0 Journal Article %A Lipton, Robert %A Mengesha, Tadele %T Representation formulas for $L^\infty $ norms of weakly convergent sequences of gradient fields in homogenization %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 1121-1146 %V 46 %N 5 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2011049/ %R 10.1051/m2an/2011049 %G en %F M2AN_2012__46_5_1121_0
Lipton, Robert; Mengesha, Tadele. Representation formulas for $L^\infty $ norms of weakly convergent sequences of gradient fields in homogenization. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 5, pp. 1121-1146. doi: 10.1051/m2an/2011049
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